Finanical instrument portfolio credit exposure evaluation

ABSTRACT

The present invention relates to a method for use in the evaluation of the credit exposure of a portfolio of financial instruments.

The present invention relates to a method for use in the evaluation ofthe credit exposure of a portfolio of financial instruments.

Financial instruments such as forwards, options and other derivativescan be highly complex, making an assessment of their future value overtime a complicated and highly specialised area.

It will be appreciated that the value of such instruments at any onefuture point in time will depend not only on the “internal” nature ofthe instrument itself, but also on “external” market factors, such asfor example future exchange rates, asset prices and interest rates.

The “credit exposure” of a portfolio of financial instruments isdefined—for a future point in time—as the statistical upper-bound(usually 97.5% or 99%) of the possible values which might be attained bythat portfolio at that future time. The construction of future valuesmay also incorporate the effect of legal contracts which reduce creditexposure in the event of client default such as collateral agreementsand close-out netting agreements.

A reliable estimate of credit exposure (statistical upper-bound offuture value) for a portfolio of such financial instruments can beinvaluable to a financial institution. In particular, a profile ofcredit exposure at many future points in time is often used whenassessing whether to enter into a financial contract with a client. Forexample, a “Tier 1” bank, such as ABN-AMRO may be considering whether totrade a particular financial instrument with a client, say Shell.ABN-AMRO need to be sure however that the addition of that financialinstrument to Shell's existing portfolio does not create inappropriatefuture credit exposures: (a) that Shell will be able to meet its worstcase obligations on the portfolio and (b) that the level of creditexposure is proportional to the risk of Shell being put intoreceivership.

In this connection, mathematical systems are currently available foranalysing a portfolio of financial instruments with a view topredicting, at a future point in time, the statistical upper-bound ofthe value of the portfolio. The statistical upper bound of the portfoliocan be seen as the worst case scenario for what could be lost, i.e. themaximum that may be lost if the counterparty cannot cover that maximumvalue due to default.

One such method for assessing the statistical upper-bound utilises theMonte Carlo technique. The method requires that each financialinstrument can be “valued” using a mathematical formula which gives thefuture value of the instrument from hypothetical future values ofunderlying market variables (asset prices, exchange rates, interestrates). The value of a portfolio at some arbitrary future date can hencebe expressed as the combination of valuation functions of theconstituent financial instruments. This results in a very largemathematical function, say P( ), which depends on a very large number ofunknown market variables.

The Monte Carlo method establishes the upper-bound of portfolio value bygenerating future random values for all the variables that the portfoliodepends upon and applying the portfolio valuation function, P( ), tothose variables. If this is done often enough (10,000 is typical) thenone can estimate the upper-bound at a given point in time by orderingthe outcomes an picking the valuation that occurs at the appropriaterank.

In other words, at each designated point in time, the method will carryout thousands of calculations to evaluate all potential values of theportfolio at that point. The reliability of the estimate will increasefor an increased number of calculations, but this can be prohibitivesince the speed of securing a result is directly related to the numberof calculations made. The Monte Carlo method is laborious and hencerelatively slow.

Since the portfolio valuation function can be so complex, there is noguaranteed shortcut for carrying out all this multi-dimensionalstatistics without simulation.

An object of the present invention is therefore to seek to provide anevaluation method that alleviates such problems of existing methods.

According to a first aspect of the present invention there is provided amethod for evaluating the credit exposure of a portfolio of one or morefinancial instruments, the method comprising:

-   -   establishing a deal object for the or each financial instrument,        the deal object comprising a representation of said financial        instrument and a valuation function for representing how the        value of the financial instrument is related to underlying        market variables.    -   establishing one or more risk factor models, the or each risk        factor model representing an underlying financial market        variable which may affect the value of one or more of said        financial instruments;    -   establishing a deal parabolic function for representing the or        each deal object valuation function, by operation of said the or        each deal object valuation function on said the or each risk        factor model to which it is sensitive, each deal parabolic        function giving its value at a particular instant under a        particular state of market risk factors;    -   summing the coefficients of each said deal parabolic function        established at a same instant from the one or more deal objects        represented in the portfolio in order to build a portfolio        parabolic function which approximates the overall portfolio        value for that instant;    -   wherein establishing each said deal parabolic function involves        evaluating a plurality of coordinates calculated from said deal        object valuation function and then applying a parabolic curve,        surface or multi-dimensional surface (manifold) to fit said        coordinates, the parabolic curve or surface then representing        that deal object valuation function.

Preferably, three or more coordinates are calculated and plotted towhich the parabolic curve or quadratic polynomial is to be applied.

The evaluation method of the present invention is not hence based aroundusing random elements, but instead simplifies the problem of valuingcomplex functions so that an answer can be determined by linear algebraand established properties of quadratic forms.

In this regard, the evaluation method of the present invention involvesthe creation of a simplified approximation of each deal object valuationfunction, a hyper-dimensional surface or manifold. This manifold ispreferably quadratic (in any one dimension only depends on threeparameters).

In preferred embodiments, the or each deal object valuation function isestablished using a deal object, each deal object operating to select anappropriate skeleton or template representation from a store of suchrepresentations of financial instruments, and to populate the skeletonor template with data which will define the nature of the financialinstrument.

Conveniently, the or each deal object further operates to identify towhich risk factor models the valuation function of that deal object issensitive.

Preferably, the or each deal object further operates to value the dealrepresented therein in relation to the risk factor models that have beenidentified as being appropriate to the valuation function of that dealobject. For a single future point in time several valuations areperformed corresponding to different state of the contingent risk factormodels.

In preferred embodiments, the deal object operates to apply an “optimalfit” parabolic curve to the coordinates evaluated from the dealvaluation function, the optimal fit curve being chosen on the basis thatit most readily passes through the coordinates plotted with minimumdeflection between coordinates. The exact position and number of thevaluations conducted depends on the type of deal object. Where threepoints per risk factor are evaluated, an exact parabola fit can beestablished. Where more than three points per risk factor are evaluatedthe technique of quadratic least squares is employed to derive theparabola coefficients.

Preferably, the coordinates to which the parabola is to be applied arechosen from each of the limits of the deal object valuation function(defined by the extremes of the relevant risk factors) and at itscentral region.

Deal objects can be of various types depending on the type of financialinstrument represented. For example a Foreign Exchange Option will berepresented by an FXOptionDeal object. This object will have parameterswhich store all the details (terms & conditions) of that particular deale.g whether it is a ‘call’ or ‘put’, whether it is bought or sold, theoption strike amount, the underlying amount, the trade date, and theexercise date.

If a portfolio contains 1000 foreign exchange options then it will berepresented by 1000 deal objects, all with different data fields, butwith the same type. The structure of the data they hold and the codethat is operable on them will be the same. But of course, portfolios arelikely to contain many types of financial instrument. In the currentsystem there are around 50 different types of deal objects. Templates orskeleton representations of such deal objects are stored in a suitablestore.

Preferably, the or each risk factor model is established using a riskfactor object, each risk factor object operating to select anappropriate skeleton or template representation from a store of suchrepresentations of risk factors and to populate the skeleton or templatewith data which will define the nature of the risk factor model.

The value of a deal at any point in time depends on certain marketvariables which are termed risk factors. For example the value of aGBP:USDOLLAR option deal is in essence determined by the behavior of (a)the GBP:USD exchange rate and (b) the USD interest rate (if we arevaluing the option in USD). The option will only be affected by othermarket variables in as much as they affect these two “underlyings”. Thesame is true of the portfolio as a whole. A counterparty portfolio mightcontain 1000 deals and its value dependent on 100 risk factors.Templates or skeleton representations for such risk factors are storedin a Risk Factor List.

Conveniently, the method further comprises compiling a database or storeof financial and market data.

Preferably, the method comprises the use of techniques of linear algebraand quadratic forms to evaluate the statistical upper-bound of theparabolic function for the overall portfolio, and hence, anapproximation for the credit exposure of the portfolio.

In preferred embodiments, said techniques comprise one or more ofaggregation, decomposition of the covariance matrix, transformation ontoorthogonal variables, the application of PCA (Principal ComponentAnalysis) on weighted factors, the transforming away of cross terms, andthe calculation of distribution moments and percentiles for theapproximation.

According to a further aspect of the present invention there is provideda system for evaluating the exposure of a portfolio of one or morefinancial instruments, the system comprising:

-   -   means for establishing a deal object for the or each financial        instrument, the deal object comprising a representation of said        financial instrument and a valuation function for representing        how the value of the financial instrument is related to        underlying market variables;    -   means for establishing one or more risk factor models, the or        each risk factor model representing an underlying financial        market risk factor which may affect the value of one or more of        said deal objects;    -   means for establishing a deal parabolic function for each deal        object, comprising means for operation of the deal object        valuation function on said the or each risk factor model to        which it is sensitive; and    -   means for summing each said deal parabolic function established        at a same instant from the deal objects represented in the        portfolio in order to build an overall portfolio value for that        instant;    -   wherein said means for establishing each said deal parabolic        function comprises means for making an approximation of said        each deal object valuation function, involving the evaluation of        a plurality of coordinates calculated from said deal object        valuation function and then applying a parabolic curve, surface,        hyper surface (manifold) to fit said coordinates, the parabolic        curve or quadratic polynomial then representing that deal        valuation function.

A system as defined above, wherein the system is a computer system.

The present invention encompasses a computer program comprising programinstructions for carrying out the above method a computer program asdefined above, the program being embodied on a record medium, computermemory, read-only memory, or electrical carrier signal.

An example of the present invention will now be described with referenceto the attached drawing, FIG. 1, which shows diagrammatically the methodaccording to the present invention.

It will be appreciated that the method that forms the present inventioncan be implemented on a computer based machine. As such the elementsrepresented in FIG. 1 correspond to computer system (a computer-basedmachine including computer hardware and software).

In this regard a computer system performing the method of the presentinvention, is used to calculate the upper-bound of the possible futurevalues of a portfolio of financial instruments. The calculation can beequated to a maximum exposure of that portfolio and hence can be used bya financial institution to evaluate whether a particular portfolio isappropriate for a particular counterparty.

As a first step in the method, predetermined mathematicalrepresentations of the financial instruments to form the portfolio arecalled up from an electronic memory store and populated with data from adatabase to thereby define the nature of each of the respectivefinancial instruments.

Hence, for each type of financial instrument to the portfolio thereexists a electronic basic skeleton or template representation. Such askeleton is matched to the financial instrument in question, and thenpopulated with data from a data base, to define the financial instrumentmore precisely. As an example, there is a skeleton representation forcertain foreign exchange related financial instruments. Therepresentation will for example have parameters which store details ofwhether the instrument is a “call” or “put”, whether it is bought orsold, a strike amount, an underlying amount, a trade date and anexercise date. Such data is populated into the skeleton function todefine the nature of the instrument.

The defined representations of the financial instruments are termed“deal objects”. The deal objects have three main functions:

The first function of each deal object is to value itself using its dealvaluation function for a given scenario of market variables. Thevaluation function is an established mathematical function which isdifferent for each deal type. The value of the function depends on thevalues of market variables represented on the risk factor objects andalso the representation of the financial instrument contained in thedeal object.

The second function of each deal object is to establish to whichexternal or market factors the valuation function of that deal object issensitive. For example, for a foreign exchange financial instrument,factors such as the USDollar/Sterling rate can have significant effectson their valuation. As such, the second function of each deal object isto identify which external factors (termed risk factors) are appropriateto the deal valuation function. In this connection, the future value ofa financial instrument(or portfolio of instruments) will be affected byone or more inherently random market variables—i.e. the risk factors.Thus when making a future valuation of a particular instrument, oneneeds to take into account the effects of such underlying factors.

To this end, risk factor objects operate within the method to evaluatethe effects of such external risk factors. The risk factor objectsoperate in a similar way to the deal objects, namely for each type ofrisk factor, such as asset prices or interest rates, there is anelectronic skeleton or template in the form of a mathematicalrepresentation. These representations are populated with data from adatabase to define more precisely what that risk factor is and therebyform one or more risk factor models. The risk factor objects are hence astatistical description of the market variable in question, say UKinterest rates, together with a particular state of that variable. The“state” of each risk factor is used throughout the method in order torepresent different future market scenarios to the deal valuationfunctions.

The third function of each deal object is to approximate its valuationfunction with a parabolic function. The parabolic function for each dealobject is determined using the deal valuation function and the relevantrisk model or models, and is represented by the formula:U( x (t), t)=a(t)+Σb _(i)(t)·x _(i)(t)+Σc _(ij)(t)·x _(i)(t)·x _(j)(t)

Where:

-   -   U(x(t),t) is the parabolic approximation of the valuation        function.    -   x(t) is multivariate normal vector of risk factors.    -   a, b, c are time-dependent constants.

In this regard, each deal object is able to select how its dealvaluation function fits a parabola in the “optimal” way such that itsbehavior at the extremes and any internal nuances are captured. Fittingthe “optimal fit” parabolic curve for a particular deal valuationinvolves establishing a limited number (e.g. three) or more coordinatesthrough actual calculation, and then fitting the optimal curve to passthrough those coordinates.

By way of a single-dimension simplified example, if we consider thepossible future values of a particular deal valuation function (d_(i))represented graphically by the following curves over time (t).

In this example if we consider this deal valuation function to besensitive to a single risk factor (RF_(j)), say interest rates. Then, ata particular instant, the valuation of d_(i) is found by the operationof d_(i) (RFj) and may take the form:

As discussed, in order to plot each point on this deal valuationfunction would involve many thousands of test cases. Since this wouldsignificantly reduce the speed of operation of the method, rather thanevaluate each point, the method makes an approximation of the valuationfunction by evaluating the deal valuation function at three or morevalues of the risk factor (RF_(i)) and then fitting the optimal parabolathrough these points, as shown by the dotted line passing throughcoordinates A, B and C.

Each deal object is able to select how many coordinates to take to makethe optimal fit parabola and is further able to choose what form ofparabola will fit in the most suitable way.

It should be borne in mind that the above example is highly simplifiedand that in practice the deal valuation functions and risk factor modelswill be multi-dimensional and highly complex, with each deal valuationfunction being sensitive to a plurality of such risk factor models.

In this connection, a more complete analysis of the fitting of theoptimal parabola (or quadratic polynomials) is given below.

Fitting the Parabolic Approximation

The key to the method is that deals of different types are able toselect the method used to construct their optimal parabolic function.This parabolic function approximates the valuation function as closelyas possible particular at the extremes of the risk factor where theupper-bound is most likely to be situated.

Non-Linear Functions

For non-linear products in the netted case, and all products which cancross the money in the non-netted case (when used without netting, thefloor of the valuation function is used), the following technique isused to fit parabolic manifolds. As already discussed, it is assumedthat each valuation function can be expressed in the form:

-   -   V(x(t),t) where x(t) is a multivariate normal distribution with        zero mean        or risk factors that are prices, the random variable used will        be x(t) used by the lognormal diffusion model such that:        x(t)≈φ(0, σ{square root}{square root over (t)})        S(t)={overscore (S)}(t)·e ^(x(t))        S(t)=S₀e^(μ·t−0.5σ) ² ^(t)

For risk factors that are interest rates the random variable used willbe x(t) used by the Hull-White mean reversion model such that:x(t)≈φ(0,{square root}{square root over ((e^(2αt)−1)/2α)})DF(t,T)=H(t,T)·e ^(J(t,T)·x(t))J(t,T)=−σ·e ^(−αt) ·B(t,T)

We seek to approximate each valuation function with a polynomial of theform:${U\left( {\underset{\_}{x},t} \right)} = {{a(t)} + {\sum\limits_{i}{{b_{i}(t)} \cdot {x_{i}(t)}}} + {\sum\limits_{i,j}{{c_{ij}(t)} \cdot {x_{i}(t)} \cdot {x_{j}(t)}}}}$

The polynomial coefficients are generated so as to create a manifoldthat intersects the valuation function at the origin, and at the singledimensional upper and lower bounds of each risk factor. This can beregarded as a ‘discretised’ Taylor expansion which tends to the Taylorcoefficients (where calculable) as σ_(i)(t) tends to zero.a(t) = V(0, t)${b_{i}(t)} = \frac{\left( {{V\left( {q_{i},t} \right)} - {V\left( {{- q_{i}},t} \right)}} \right)}{2 \cdot {\sigma_{i}(t)} \cdot Q_{a}}$${c_{ii}(t)} = \frac{\left( {{V\left( {q_{i},t} \right)} + {V\left( {{- q_{i}},t} \right)} - {2 \cdot {V\left( {0,t} \right)}}} \right)}{2 \cdot \left( {{\sigma_{i}(t)} \cdot Q_{a}} \right)^{2}}$

Where:

-   -   q_(i) is a vector with the i^(th) element set to σ_(i)(t)·Q_(α)        and all others zero    -   Q_(α) is the normal quantile at the required level of confidence

When second-order cross terms are required, they are given (for i≠j) by:${c_{ij}(t)} = \frac{\left( {{V\left( {{{+ q_{i}} + q_{j}},t} \right)} + {V\left( {{{- q_{i}} - q_{j}},t} \right)} - {V\left( {{{+ q_{i}} - q_{j}},t} \right)} - {V\left( {{{- q_{i}} + q_{j}},t} \right)}} \right)}{2 \cdot {\sigma_{i}(t)} \cdot {\sigma_{j}(t)} \cdot Q_{a}^{2}}$

Where more than three points are evaluated for each risk factor, theprocess of quadratic least squares is used to fix the quadraticcoefficients.

Linear Functions

For linear valuation functions in the netted case, such as forwards andvanilla interest rate swaps, it is possible to generate the coefficientsof the parabolic manifold directly from the valuation functionsthemselves. Examples of direct derivation of coefficients are givenbelow for a Forward Rate Agreement, and Equity Forward.

The Forward Rate Agreement (FRA) can be written as a parabolic surfaceof two risk factors with coefficients:α=P·{overscore (X)}(t){H(t,T)−K·H(t,T+m)}b ₁ =P·{overscore (X)}(t){H(t,T)−K·H(t,T+m)}c ₁₁ =P·{overscore (X)}(t)·{H(t,T)−K·H(t,T+m)}/2b ₂ =P·{overscore (X)}(t)·(H(t,T)J(t,T)−K·H(t,T+m)J(t,T+m))c ₂₂ =P·{overscore (X)}(t)·(H(t,T)J ²(t,T)−K·H(t,T+m)J ²(t,T+m))/2c ₁₂ =P·{overscore (X)}(t)·(H(t,T)J({overscore(t)},T)−K·H(t,T+m)J(t,T+m))K=(1+(m/360)·FRA)where x₁ is the risk factor for the exchange rate to base currency, andx₂ is the risk factor for the FRA interest rate.

Similarly, the Equity Forward can be expressed in terms of the riskfactors for equity price (x₁) deal exchange rate (x₂) and deal interestrate (x₃) using the following coefficients:α={overscore (S)}(t)·e ^(−D(T−t)) −S _(—) FWD·H(t,T)·{overscore (X)}(t)b ₁ ={overscore (S)}(t)·e ^(−D(T−t))b ₂ =−S _(—) FWD·H(t,T)·{overscore (X)}(t)b ₃ =−S _(—) FWD·H(t,T)·{overscore (X)}(t)·J(t,T)c ₁₁ ={overscore (S)}(t)·e ^(−D(T−t))/2c ₂₂ =−S _(—) FWD·H(t,T)·{overscore (X)}(t)/2c ₃₃ =−S _(—) FWD·H(t,T)·{overscore (X)}(t)·J ²(t,T)/2c ₂₃ =S _(—) FWD·H(t,T)·{overscore (X)}(t)·J(t,T)Variations Some deal functions work well with a hybrid of thediscretised and Taylor methods. For example, the value of a Europeanequity call option is given by:V(t)=X(t)[S _(d)(t)·e ^(−D(T−t)·) N(d ₁)−K·DF _(d)(t,T)·N(d ₂)]

The value of the option in the deal's domestic currency is theexpression within the square brackets which becomes increasinglynon-linear as deal approaches maturity. However the exchange rate whichconverts the function back to base currency can be modelled with aTaylor expansion:X(t)=α+bx(t)+cx ²(t)α=X(0)·e ^((μ) ² ^(−σ) ² ² ^(/2)t)b=X(0)·e ^((μ) ² ^(−σ) ² ² ^(/2)t)c=X(0)·e ^((μ) ² ^(−σ) ² ² ^(/2)t)/2

The non-linearity of the option equation in base currency can thus bemodelled by taking the product of the two expansions resulting in acloser fit polynomial.

This third operation of each deal object therefore builds parabolicfunctions which represent valuations of each deal function taking intoaccount the risk factors to which that deal function is sensitive.

Upper-Bound of the Parabolic Manifold

In order to evaluate the upper-bound of the portfolio the method usesthe upper bound of the parabolic manifold of the portfolio as anapproximation. The portfolio parabolic manifold is built under acalculation object, which in essence involves adding the parabolicfunctions of each of the deal function valuations built in the dealobjects discussed above.

Hence if there are 3 deals, the parabolic functions of these are addedto arrive at a total. In practice in building the parabolic manifold,the coefficients of corresponding terms are added such that for eachdeal valuation function d_(i) $\begin{matrix}d_{1} & {V_{1} = {a_{1} + {b_{1.}x} + {c_{1.}x^{2}}}} \\d_{2} & {V_{2} = {a_{2} + {b_{2.}x} + {c_{2.}x^{2}}}} \\d_{1} & {V_{3} = {a_{3} + {b_{3.}x} + {c_{3.}x^{2}}}} \\{SUM} & {V_{3} = {A + {B.x} + {C.x^{2}}}}\end{matrix}$

Where A=a₁+a₂+a₃

-   -   B=b₁+b₂+b₃    -   C=c₁+c₂+c₃

This parabolic manifold will relate to a highly complex surface. Theupper bound of this manifold is then evaluated by certain mathematicalmethods, such as aggregation, decomposition of the covariance matrix,transformation onto orthogonal variables, the application of PCA(Principal Component Analysis) on weighted factors, the transformingaway of cross terms, and the calculation of distribution moments andpercentiles for the approximation. These methods are set out below ingreater detail in the context of the process of establishing anupper-bound for the parabolic manifold at a given point in time.

Step 1: Portfolio Parabolic Manifold

As already discussed, the portfolio is approximated by a parabolicmanifold of normal variates with constants A(t)m B_(i)(t) and C_(ij)(t)of the form: $\begin{matrix}\begin{matrix}{{\overset{\_}{\Pi}\left( {\underset{\_}{x},t} \right)} = {{A(t)} + {\sum\limits_{i}{{B_{i}(t)} \cdot {x_{i}(t)}}} + {\sum\limits_{i,j}{{C_{ij}(t)} \cdot {x_{i}(t)} \cdot {x_{j}(t)}}}}} \\{{A(t)} = {\sum\limits_{k}{a^{k}(t)}}} \\{{B_{i}(t)} = {\sum\limits_{k}{b_{i}^{k}(t)}}} \\{{C_{ij}(t)} = {\sum\limits_{k}{c_{ij}^{k}(t)}}}\end{matrix} & (0.1)\end{matrix}$Step 2: Decompose the Covariance Matrix

In preparation for transforming onto an orthogonal set of risk factors,we find the ‘square root’ (Cholesky factorisation) of the covariancematrix COV for the stochastic variables x_(i). This yields a matrix Dsuch that:COV=D·D ^(T)

Note that the Cholesky factorisation of a covariance matrix can bederived from the decomposition of the correlation matrix as follows. Letthe matrix A=[a_(ij)] be the decomposed correlation matrix such that:AA^(T)=[ρ_(ij)]

Then the decomposition of the covariance matrix is given by [σ_(i)a_(ij)] since: $\begin{matrix}{{\left\lbrack {a_{ij}\sigma_{i}} \right\rbrack\left\lbrack {a_{ij}\sigma_{i}} \right\rbrack}^{T} = {{{diag}\left( \underset{\_}{\sigma} \right)}{A \cdot \left\lbrack {{{diag}\left( \underset{\_}{\sigma} \right)}A} \right\rbrack^{T}}}} \\{= {{{diag}\left( \underset{\_}{\sigma} \right)}{A \cdot A^{T}}{{diag}\left( \underset{\_}{\sigma} \right)}^{T}}} \\{= {{{{diag}\left( \underset{\_}{\sigma} \right)}\left\lbrack \rho_{ij} \right\rbrack}{{diag}\left( \underset{\_}{\sigma} \right)}}} \\{= {COV}}\end{matrix}$Step 3: Transform Onto Orthogonal Variables

We make the transformation onto the orthogonal vector, y, defined by:y(t)=D ⁻¹ ·x (t) or x(t)=D·y (t)

Then y(t) is multivariate normal with zero mean and unit covariancematrix. We may now write the portfolio approximation as:{overscore (Π)}( y, t)=A(t)+{tilde over (B)} ^(T) ·y (t)+ y^(T)(t)·{tilde over (C)}·y( t)where:{tilde over (B)}=D ^(T) ·B( t){tilde over (C)}= D ^(T) ·C(t)·Dy( t)−Φ(0,I)Step 4: Perform PCA on Weighted Factors

At this stage the dimensionality of the calculation may be quitehigh—particularly if the portfolio contains a large number of equity orcommodity derivatives. As long as these outlying assets have beencorrelated reasonably to more dominant risk factors (such as stockindices, the price of their trading currency etc), then movements inthese variables is encapsulated in the dominant terms of the neworthogonal factors, y.

To establish the principal components of the y factors, we take theabsolute size of each first order term as a measure of its significanceto the portfolio value at that timestep.

Suppose $\underset{\_}{\overset{\sim}{B}} = \begin{pmatrix}{\overset{\sim}{b}}_{1} \\{\overset{\sim}{b}}_{2} \\\ldots \\{\overset{\sim}{b}}_{n}\end{pmatrix}$then we order {tilde over (b)}_(i) such that |{tilde over (b)}₁|≦|{tildeover (b)}₂|≦. . . ≦|{tilde over (b)}_(n)|

In parallel with this process, for each switch in position of theelements of B, we interchange both rows and columns of C. Thus if weswitch elements {tilde over (b)}_(i) and {tilde over (b)}_(j) we alsointerchange i^(th) and j^(th) column and row of C. (This maintainsconsistency between the implicit permutation applied to the factorvector).

The elements of the orthogonal vector y are now in principal componentorder, thus in order to reduce the dimensionality of the calculation thefirst k elements may be used and the others set to zero. Empirically,the calculation functions satisfactorily with the number of principalcomponents set between 10 and 20.

Step 5: Transform Away the Cross-Terms

The elements of the C matrix still contain non-diagonal elements, so wemake one last transformation to simplify the representation. Note thatafter PCA has been applied, this step is performed on a much reducedmatrix, which greatly increases speed of the o(n³) diagonalisationalgorithm. We determine an orthogonal matrix E, and a diagonal matrixĈ(t)such thatĈ(t)=E ^(T) ·{overscore (C)}(t)·E and E·E ^(T) = I

Then make the transformation onto the vector, z, defined by:z (t)=E^(T) ·y (t) or y (t)=E·z (t)

Note that since E·E^(T)=I, the vector z(t) is still (a) orthogonal and(b) multivariate normal with zero mean and unit covariance matrix. Wemay now write the portfolio approximation as:{overscore (Π)}( z (t),t)=A(t)+{circumflex over (B)}(t)^(T) ·z (t)+ z^(T)(t)·Ĉ(t)· z (t)where:{circumflex over (B)}=E ^(T) ·{tilde over (B)}(t) Ĉ=E ^(T) ·{tilde over(C)}(t)·E z (t)−Φ(0,I)

We have thus diagonalised both the covariance matrix and the portfolio‘Gamma’ matrix C (see also Rouvinez, 1997).

Step 6: Distribution Moments

We have transformed the representation of the portfolio valuationfunction into a quadratic form of independent unit normal variables.Leaving aside the constant term, A(t), which will be added to thepercentile in the last stage of the calculation, we define the variableX as a quadratic form in normal variables:X=b′y+y′Cy

A standard application of moment generating functions (see Johnson &Kotz, 1995) yields the following expressions for the moments (orcumulants) of X: $\begin{matrix}{{Mean}\text{:}} & {\mu = {{E(X)} = {{tr}(C)}}} \\{{Variance}\text{:}} & {\sigma^{2} = {{E\left( \left( {X - \mu} \right)^{2} \right)} = {{2{{tr}\left( C^{2} \right)}} + {b^{\prime}b}}}} \\{{Skew}\text{:}} & {ɛ = {\frac{E\left( \left( {X - \mu} \right)^{3} \right)}{\sigma^{3}} = \frac{{8{{tr}\left( C^{3} \right)}} + {6b^{\prime}{Cb}}}{\sigma^{3}}}} \\{{Kurtosis}\text{:}} & {\kappa = {{\frac{E\left( \left( {X - \mu} \right)^{4} \right)}{\sigma^{4}} - 3} = \frac{{48{{tr}\left( C^{4} \right)}} + {48b^{\prime}C^{2}b}}{\sigma^{4}}}}\end{matrix}$

More generally, the r^(th) cumulant is given by:$\kappa_{r} = {\frac{E\left( \left( {X - \mu} \right)^{r} \right)}{\sigma^{r}} = \frac{{2^{({r - 1})}{\left( {r - 1} \right)!}{{tr}(C)}^{r}} + {{r!}2^{({r - 3})}b^{\prime}C^{({r - 2})}b}}{\sigma^{r}}}$

Although these expressions are valid for quadratic forms containingnon-diagonal elements in the C matrix, the removal of off-diagonalelements greatly speeds up the calculation of moments because (a) thetrace (tr(C)) of a diagonal matrix is just the sum of the diagonalelements and (b) multiplication of diagonal matrices is an o(n) ratherthan an o(n³)operation.

Step 7: Percentiles for the Approximation

From the previous step, the portfolio approximation can be written as:{overscore (Π)}(t)=A(t)+X(t)

Where X(t) is a quadratic form of normal variables where all thecumulants are known. The Cornish Fisher moment correction technique isused to estimate the percentile (i.e. the exposure number) of thisdistribution. The q-percentile of a compound distribution of X involvingfour moments is given by:$X_{a} = {\mu + {\sigma\left\{ {q + {\frac{1}{6}\left( {q^{2} - 1} \right)ɛ} + {\frac{1}{24}\left( {q^{3} - {3q}} \right)\kappa} - {\frac{1}{36}\left( {{2q^{3}} - {5q}} \right)ɛ^{2}} + \ldots} \right\}}}$where:

-   -   q is the percentile of the unit normal distribution.    -   μ, σ, ε, κ are the mean, standard deviation, skew and kurtosis        of X

The q-percentile for a distribution using the first six moments is givenby: $X_{a} = {\mu + {\sigma{\left\lceil \begin{matrix}{q + {\frac{1}{6}\left( {q^{2} - 1} \right)\kappa_{3}} + {\frac{1}{24}\left( {q^{3} - {3q}} \right)\kappa_{4}} - {\frac{1}{36}\left( {{2q^{3}} - {5q}} \right)\kappa_{3}^{2}} +} \\{{\frac{1}{120}\left( {q^{4} - {6q^{2}} + 3} \right)\kappa_{3}} - {\frac{1}{24}\left( {q^{4} - {5q^{2}} + 2} \right)\kappa_{3}\kappa_{4}} +} \\{{\frac{1}{324}\left( {{12q^{4}} - {53q^{2}} + 17} \right)\kappa_{3}^{3}} + {\frac{1}{720}\left( {q^{5} - {10q^{3}} + {15q}} \right)\kappa_{6}} -} \\{{\frac{1}{180}\left( {{2q^{5}} - {17q^{3}} + {21q}} \right)\kappa_{3}\kappa_{5}} - {\frac{1}{384}\left( {{3q^{5}} - {24q^{3}} + {29q}} \right)}} \\{\kappa_{4}^{2} + {\frac{1}{288}\left( {{14q^{5}} - {103q^{3}} + {107q}} \right)\kappa_{3}^{2}\kappa_{4}} -} \\{{\frac{1}{7776}\left( {{252q^{5}} - {1688q^{3}} + {1511q}} \right)\kappa_{3}^{4}} + \ldots}\end{matrix} \right\rceil.}}}$

The exposure of the portfolio is finally given by:{overscore (Π)}⁺(t)=A(t)+X _(α)(t)

Note that this method functions for both netted and non-nettedaggregations since these are determined by the first stage of valuationfunction approximation.

In a final operation, undertaken by the Portfolio Model object, theenhanced parabolic manifold, formed in the calculation object, isoperated, i.e. it provides values for the deal valuation functionstaking into account the controlled perturbations of the risk factors inthe risk factor objects.

The present invention is not to be limited in scope by the specificembodiment described herein. Indeed, various modifications of theinvention will become apparent to those skilled in the art from theforegoing description and accompanying figure. Such modifications areintended to fall within the scope of the appended claims.

1. A method for evaluating the credit exposure of a portfolio of one or more financial instruments, the method comprising: establishing a deal object for the or each financial instrument, the deal object comprising a representation of said financial instrument and a valuation function for representing how the value of the financial instrument is related to underlying market variables; establishing one or more risk factor models, the or each risk factor model representing an underlying financial market variable which may affect the value of one or more of said financial instruments; establishing a deal parabolic function for representing the or each deal object valuation function, by operation of said the or each deal object valuation function on said the or each risk factor model to which it is sensitive; and summing the coefficients of each said deal parabolic function established at a same instant from the one or more deal objects represented in the portfolio in order to build a portfolio parabolic function which approximates the overall portfolio value for that instant; wherein establishing each said deal parabolic function involves evaluating a plurality of coordinates calculated from said deal object valuation function and then applying a parabolic curve, surface or multi-dimensional surface (manifold) to fit said coordinates, the parabolic curve or surface then representing that deal object valuation function.
 2. A method according to claim 1, wherein three or more coordinates are calculated.
 3. A method according to claim 1 wherein the or each deal object valuation function is established using a deal object, each deal object operating to select an appropriate skeleton or template representation from a store of such representations of financial instruments, and to populate the skeleton or template with data which will define the nature of the financial instrument.
 4. A method according to claim 1, wherein the or each deal object further operates to identify to which risk factor models the valuation function of that deal object is sensitive.
 5. A method according claim 1, wherein the or each deal object further operates to value the deal represented therein in relation to the risk factor models that have been identified as being appropriate to the valuation function of that deal object.
 6. A method according claim 1, wherein the deal object operates to apply an “optimal fit” parabolic curve to the coordinates evaluated from the deal object valuation function, the optimal fit curve being chosen on the basis that it most readily passes through the coordinates plotted with minimum deflection between coordinates.
 7. A method according to claim 6 wherein when more than three coordinates per risk factor model are evaluated, the technique of quadratic least squares is employed to derive the parabola coefficients.
 8. A method according to claim 1 wherein the coordinates to which the parabola is to be applied are chosen from each of the limits of the deal object valuation function (defined by the extremes of the relevant risk factors) and at its central region.
 9. A method according to claim 1, wherein templates or skeleton representations for each said deal object are stored in a suitable store.
 10. A method according claim 1, wherein the or each risk factor model is established using a risk factor object, each risk factor object operating to select an appropriate skeleton or template representation from a store of such representations of risk factors and to populate the skeleton or template with data which will define the nature of the risk factor model.
 11. A method according to claim 1, wherein the method further comprises compiling a database or store of financial and market data.
 12. A method according to claim i, fuirther comprising the use of techniques of linear algebra and quadratic forms to evaluate the statistical upper-bound of the portfolio parabolic function, and hence, an approximation for the credit exposure of the portfolio.
 13. A method according to claim 12, wherein said techniques comprise one or more of aggregation, decomposition of the covariance matrix, transformation onto orthogonal variables, the application of PCA (Principal Component Analysis) on weighted factors, the transforming away of cross terms, and the calculation of distribution moments and percentiles for the approximation.
 14. A computer program comprising program instructions for carrying out the method according to claim
 1. 15. A computer program according to claim 14, the program being embodied on a record medium, computer memory, read-only memory, or electrical carrier signal.
 16. A method according to claim 1, wherein the method is implemented on a computer.
 17. A system for evaluating the exposure of a portfolio of one or more financial instruments, the system comprising: means for establishing a deal object for the or each financial instrument, the deal object comprising a representation of said financial instrument and a valuation function for representing how the value of the financial instrument is related to underlying market variables; means for establishing one or more risk factor models, the or each risk factor model representing an underlying financial market risk factor which may affect the value of one or more of said deal objects; means for establishing a deal parabolic function for each deal object, comprising means for operation of the deal object valuation function on said the or each risk factor model to which it is sensitive; and means for summing each said deal parabolic function established at a same instant from the deal objects represented in the portfolio in order to build an overall portfolio value for that instant; wherein said means for establishing each said deal parabolic function comprises means for making an approximation of said each deal object valuation function, involving the evaluation of a plurality of coordinates calculated from said deal object valuation function and then applying a parabolic curve, surface, hyper surface
 18. A system according to claim 17, wherein the system is a computer system. 